2 Answers
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I do remember several scenes (quick ones) and off-the cuff explanations that they were receiving supplies through the man-hole that Blake's partner emerges out of near the climax as well as through the destroyed football stadium (at one point you see a whole pallet of supplies being hoisted down from...somewhere).

While vague, the audience is expected to assume the supplies they were receiving were enough not only for the citizens of Gotham, but enough to deposit down to the police's prison underground, allowing them proper sustenance.

The only logical explanation that Bane kept them alive aside from arrogance that no one would attempt to free them; is for Bruce or even Gordon to know the full extent of their failure to the city, having the police suffer down in the confined depths. If memory serves, however, Bruce doesn't know about the plot to trap the police until he finally returns to the city, so it is more likely this is Gordon's punishment.

It might be an attempt to humiliate them: it seemed to me that Bane tried to reverse everything(the rich living on the streets, the righteous being judged).

So in this situation, having the police trapped in a hole(maybe analogy to the "hole"/isolation where prisoners are sent for very bad behavior) and fed by the prisoners, might make sense. This would also explain why they were shaven: they were offered the same treatment criminals in a prison would get.

Still, killing all the policemen at once(crushed like bugs) would have been, in my opinion more tragic and a lot safer for him.

But, as we have seen, Bane has a sense of drama, all his actions have to be somehow theatrical(he only blows up the stadium after the anthem is sung, keeps the bomb in a truck and has it driving through the whole city, in order to remind the people of his power and the examples could go on), so aspects like safety or big sudden tragedies are not very familiar to Bane. He said it himself that he finds it more enjoyable to let the people have a hope and then crush it, repeating the process until the citizens are truly broken(the same strategy is he trying to apply to Batman).